I'll attempt to answer your question by misinterpreting it.

As pointed out in the comments, computing elliptic integrals is not going to be easy. But what if you wanted to find $n$ points arranged around an ellipse which form the vertices of a regular polygon? Now the answer to the question is given by a real algebraic variety. It's possible that this question may be computationally more tractable.

There are $2n$ variables $(x_i,y_i)$, $i=1,\ldots,n$, and $2n-1$ equations: $$ \frac{x_i^2}{a^2}+\frac{y_i^2}{b^2}=1,\ i=1,\ldots, n,$$ $$(x_i-x_{i+1})^2+(y_i-y_{i+1})^2=(x_{i+1}-x_{i+2})^2+(y_{i+1}-y_{i+2})^2, i=1,\ldots, n-1,$$ indices taken $(\mod n)$.

Also, for geometric reasons, one expects $n-1$ components to this variety, each of which is a circle. For example, if $n=5$, one would expect two solutions which are oriented in different directions, and two solutions which are star shaped. As one moves around the circle, the solution should move around.

Thus, one expects this variety to be a complete intersection defined by quadratic equations. There are methods from algebraic geometry to find solutions to such equations. There are versions of Newton's recursion which may be effective for finding a numerical solution. The dihedral symmetry might further constrain the solutions. Maybe someone could point you to some references if this sort of solution would suffice for your application?

I'll attempt to answer your question by misinterpreting it.

As pointed out in the comments, computing elliptic integrals is not going to be easy. But what if you wanted to find $n$ points arranged around an ellipse which form the vertices of an equilateral polygon? Now the answer to the question is given by a real algebraic variety. It's possible that this question may be computationally more tractable.

There are $2n$ variables $(x_i,y_i)$, $i=1,\ldots,n$, and $2n-1$ equations: $$ \frac{x_i^2}{a^2}+\frac{y_i^2}{b^2}=1,\ i=1,\ldots, n,$$ $$(x_i-x_{i+1})^2+(y_i-y_{i+1})^2=(x_{i+1}-x_{i+2})^2+(y_{i+1}-y_{i+2})^2, i=1,\ldots, n-1,$$ indices taken $(\mod n)$.

Also, for geometric reasons, one expects $n-1$ components to this variety, each of which is a circle. For example, if $n=5$, one would expect two solutions which are oriented in different directions, and two solutions which are star shaped. As one moves around the circle, the solution should move around.

Thus, one expects this variety to be a complete intersection defined by quadratic equations. There are methods from algebraic geometry to find solutions to such equations. There are versions of Newton's recursion which may be effective for finding a numerical solution. The dihedral symmetry might further constrain the solutions. Maybe someone could point you to some references if this sort of solution would suffice for your application?